This is the fourth in a series of papers studying the astrophysics and cosmology of massive, dynamically relaxed galaxy clusters. Here, we use measurements of weak gravitational lensing from the Weighing the Giants project to calibrate Chandra X-ray measurements of total mass that rely on the assumption of hydrostatic equilibrium. This comparison of X-ray and lensing masses measures the combined bias of X-ray hydrostatic masses from both astrophysical and instrumental sources. While we cannot disentangle the two sources of bias, only the combined bias is relevant for calibrating cosmological measurements using relaxed clusters. Assuming a fixed cosmology, and within a characteristic radius (r2500) determined from the X-ray data, we measure a lensing to X-ray mass ratio of 0.96 ± 9% (stat) ± 9% (sys). We find no significant trends of this ratio with mass, redshift or the morphological indicators used to select the sample. Our results imply that any departures from hydrostatic equilibrium at these radii are offset by calibration errors of comparable magnitude, with large departures of tens-of-percent unlikely. In addition, we find a mean concentration of the sample measured from lensing data of c200 = 3.0+4.4–1.8. In conclusion, anticipated short-term improvements in lensing systematics, and a modest expansionmore » of the relaxed lensing sample, can easily increase the measurement precision by 30–50%, leading to similar improvements in cosmological constraints that employ X-ray hydrostatic mass estimates, such as on Ωm from the cluster gas mass fraction.« less